Mathematical Opera – Strulovitz

Overview

This is my attempt to make an opera about the Millennium Prize Problems, if you solve any of these problems you get a prize of one million dollars from the Clay Mathematics Institute!

https://en.wikipedia.org/wiki/Millennium_Prize_Problems

So the opera will have seven acts because there are seven such problems, and in each problem the characters will use a time machine to travel to a different period in history, and they will meet with the smartest mathematical genius of that period, and he will help them to solve one of the problems.

The music of the singing parts will be based on the equations of the title problem of that act. for example in the part about the Riemann hypothesis

https://en.wikipedia.org/wiki/Riemann_hypothesis

the equation will be the Riemann zeta function.

https://en.wikipedia.org/wiki/Riemann_zeta_function

basically i will take the pictures in these Wikipedia pages and try to convey them in various ways through music.

The most obvious way is to treat the rise and fall of a graph

https://en.wikipedia.org/wiki/Riemann_hypothesis#/media/File:RiemannCriticalLine.svg

like the rise and fall of the notes on the staff

https://en.wikipedia.org/wiki/Staff_(music)

But maybe I will think of additional ways.

(As you can see this part is similar to my Sonifiquation project)

Main Plot

Remember how Doctor Who always have a woman sidekick? So I’m kidnapping my mythological ex (i apologize that it’s not educational, anyway it’s totally fictional) and our “Tardis” like time machine basically protects the time line inside it, but all the rest of the universe’s time flows in reverse. So outside everything is like a movie that is played backwards. So people are undone (including her kids and husband) and there are very few “stopping stations” that I was able to program into the machine before the process started. and i chose these as the times of those few mathematical geniuses that we’re going to meet.

So time flows in all the world backwards (but inside the time capsule we are protected and time is normal), except for the seven “stopping stations” when time in all the world behaves normally for 24 hours, and then we need to escape to the time capsule again or else we become undone.

Eventually my mythological ex comes up with a way to stop the time machine and restore the time direction back to normal (forward). By then we have returned to the days of the first homo-sapiens humans, and we both start humanity over as the new Adam and Eve! Oh and each time the time machine is working there’s the music of “Ride On Time” the song by Black Box (like instead of the Doctor Who Theme Tune).

Mathematicians (later i will assign them to the 7 problems)

Greece (geometry)
Pythagoras (circa 570-495BC)
Euclid (circa 365-275 BC)
Archimedes (circa 287–212 BC)

India (algebra)
Brahmagupta (c. 598 – c. 668 CE)
Aryabhata (476–550 CE)

England (calculus)
Newton (1642–1727)
Leibniz (1646-1716)

Russia (analysis)
Euler (1707-1783)

France (more analysis)
Lagrange (1736–1813) calculus of variations
Laplace (1749–1827) mathematical astronomy
Legendre (1752–1833)mathematical physics

thank you to Matthew Rave from Many Worlds Theory:
Lagrange, Laplace, and Legendre: which one is which?
https://manyworldstheory.com/2014/11/24/lagrange-laplace-and-legendre-which-one-is-which/

Lagrange: the beauty of math; reformulated mechanics in the Mécanique analytique

Laplace: math as a tool; Newtonian mechanics reaches its zenith in Mécanique céleste; probability theory

Legendre: the creepy looking elliptic integral guy

Germany I ( abstract algebra )
Gauss (1777-1855) everything
Abel (1802–1829) modern algebra
Jacobi (1804–1851) differential equations
Hamilton (1805–1865) mathematical physics
Galois (1811–1832) group theory
Riemann (1826–1866) differential geometry

Germany II (more abstract algebra)
Cantor (1845-1918) infinite sets
Poincaré (1854–1912) everything
Hilbert (1862 – 1943) modern algebra
Noether (1882–1935) modern algebra
Ramanujan (1887–1920) number theory

thank you to Lê Nguyên Hoang from Lê’s Blog
Rigor and Intuition in Mathematics
https://lenhoang.wordpress.com/2013/08/01/rigor-and-intuition-in-mathematics/

A hundred years, two giants of mathematics debated the role of rigor and intuition in mathematics. While the mighty David Hilbert was praising rigorous foundations for the field, the renowned Henri Poincaré was arguing that mathematics was a matter of intuitions.

Reversing Time

In the extremely interesting video by Varitasium about pilot waves:

Is This What Quantum Mechanics Looks Like?

you can see in the last few seconds of the video (starting in minute 6:48), that TIME CAN BE REVERSED!!!

how can we make the droplet land on the back side of the wave? I think it’s enough if we delay the droplet for the right amount of time in the air, then it will land on the back side of the wave. Why is that? because normally the droplet moves forward, so it lands on the forward side of the wave. So that means the wave is moving independently below the droplet while the droplet is in the air (otherwise the droplet will not land in a wave at all, it would have just created a center of a new wave). So if we stop the forward movement of the droplet for a precise fraction of time, let’s call this momentary pause “Switch Direction” (like that special card in the Israeli card game “Taki” ha ha).

I guess it’s half of the time it takes from one bounce to the next. then the wave will not know about this, so it will move forward, and the droplet will stay back, and the droplet will land on the back side of the wave and from that moment on, time is going backwards!

The “Switch Direction” pause is the same length for all the drops. because the time that the drops remain in the air is the same for all the drops. you can see that all the droplets are synchronized to the same frequency in the beginning of the video, when Derek Muller makes big droplets which move a little up and down, and small droplets which move a lot up and down.

it’s similar to the fact that the length of the string of the pendulum dictates dictates the frequency and not the size of the weight. a big weight will move a little left and right, and a small weight will move a lot left and right, but a each cycle will take the same amount of time.

OK so how do we make the “Switch Direction” pause all over the area where all the droplets are jumping, and how do we make this pause at exactly the same time?

for example in Veritasium’s example he uses Silicone Oil

https://en.wikipedia.org/wiki/Silicone_oil

Which is an electric insulator and non flammable, so we could ionize the air above the droplets which will cause the top of the droplets to become ionized. the big droplets will accumulate big ionization, and the small drops a little ionization. oil droplets can be ionized see also in this famous experiment

Oil drop experiment of Robert A. Millikan

https://en.wikipedia.org/wiki/Oil_drop_experiment

So like in the experiment we put the whole thing between two electrodes and then we switch on the current for the exact duration that the “Switch Direction” pause should be.

Or another method, just like Veritasium uses a speaker to vibrate the liquid from below, with pressurized air (the sound waves that push from below) so we can use for a very short duration (again the “Switch Direction” pause duration) a speaker from below that will suck the air and make de-pressurization that will pull the droplets a little upwards. The little droplets by a lot and the bigger droplets by a little as it should be.

So our “time machine” is something that protects whomever is inside it from this time-flowing-backwards that happens everywhere else (outside the time machine). like in the example of the petri-dish it would be some roof on some droplets. In reality the time machine needs to protect all the particles that are within its walls.

If you want to reverse time, you have to do it at all points at once

If you fall into a black hole and come out of a white hole on the other side

And they are both connected in a wormhole

On one side, the black hole takes us apart bit by bit

But on the other side, the white hole puts us back together bit by bit

If every bit that goes in goes through and comes out, then there is no loss of information

If the diameter of the wormhole is just such that the component that maintains the direction of time (the bubble jumping on the oil) slows down just enough to reverse the direction of time:

here in this video you can see how to slow down (like the marble that is doing many fast rounds at close to the bottom of the sink, it takes it much more time to fall down. (just like when a coin wobbles long time before it lays flat on the table and comes to rest).

“Marble Black Hole” – Hyperbolic Gravitational Funnel Marble Run

Jelle’s Marble Runs

And then what comes out is an exact copy of us but made of antimatter! That is, we are ordinary matter moving backwards in time! (According to Wheeler and Feynman) !

https://en.wikipedia.org/wiki/One-electron_universe

Feynman was struck by Wheeler’s insight that antiparticles could be represented by reversed world lines, and credits this to Wheeler, saying in his Nobel speech:

I did not take the idea that all the electrons were the same one from [Wheeler] as seriously as I took the observation that positrons could simply be represented as electrons going from the future to the past in a back section of their world lines. That, I stole![1]

Feynman later proposed this interpretation of the positron as an electron moving backward in time in his 1949 paper “The Theory of Positrons”. Yoichiro Nambu later applied it to all production and annihilation of particle-antiparticle pairs, stating that “the eventual creation and annihilation of pairs that may occur now and then, is no creation nor annihilation, but only a change of directions of moving particles, from past to future, or from future to past.”Then all we need to do is avoid contact with normal matter (because contact causes an explosion and will kill us) and wait each time for the time we want to travel back in time

(It is important to choose an area completely free of normal space matter, or even to “clean” an area, like the “sparks” that shoot in all directions before a space shuttle launches, so that there are no flammable gases on the launch pad that will cause an explosion; in our case, shoot an antimatter particle so that if there is normal matter, if they collide with it and explode with it, a small explosion will occur and they will disappear).

If we want to travel back 100 years and meet Ramanujan for example, then we need to go through the artificial wormhole into a state of antimatter, and then exist like that for 100 years

In order not to need food, drink and oxygen, we “freeze” ourselves for this period of time (Suspended animation Cryonics like wood frog)

Then when the computer decided it was time to get up, the computer with the help of robots (which we created in advance and have been waiting for the computer’s instruction all this time) creates a reverse initiated wormhole for us (which turns us into normal matter with the normal direction of time) and that’s how we went back in time to the desired year!

Birch and Swinnerton-Dyer conjecture. level: undergraduate. ChatGPT o1.

Overview


The Birch and Swinnerton-Dyer conjecture (often shortened to "BSD") is one of the great unsolved problems in modern number theory. It revolves around the study of , which are special types of polynomial equations that have a remarkable structure. Although these curves look simple at first glance, they lie at the heart of deep questions about rational points (solutions) and the nature of certain special functions called  .

1. Elliptic Curves and Rational Points

1. Definition
An   over the rational numbers  can be written (after suitable transformations) in the form



with , and the condition that the right-hand side polynomial has no repeated roots (this ensures "smoothness", i.e., no cusps or self-intersections).

2. Rational Points
A  on  is a solution  where both  and  are rational numbers. Surprisingly, the set of all rational points on  can be given the structure of an  under a special geometric addition law.

3. Finite vs. Infinite Solutions
Some elliptic curves have only finitely many rational points, whereas others can have infinitely many. The  of an elliptic curve is, roughly, the number of "independent directions" along which you can find infinitely many rational points.

2. The L-Function

1. Intuitive Idea
For an elliptic curve , mathematicians construct an  (or ) , which is a special function defined using local information about  at each prime . It is somewhat analogous to how the Riemann zeta function  encodes information about the primes in its definition.

2. Connection to the Curve
The conjecture says that the behavior of this L-series at a particular point  tells us crucial information about the rank of the elliptic curve. Specifically, the number of times  vanishes at  (the "order of the zero") is predicted to match exactly the rank of the curve.

3. Statement of the Conjecture

Birch and Swinnerton-Dyer Conjecture (Informal Version)

    • If , then the rank of  is . This means the elliptic curve has only finitely many rational points.
    • If  has a zero of order  at , then the rank of  is . This means there are infinitely many rational points arranged in  independent "directions".


In short, the deeper the zero at , the higher the rank of the curve .

4. Why It Matters

1. Unifying Number Theory and Geometry
Elliptic curves sit at the intersection of number theory, algebraic geometry, and complex analysis. The BSD conjecture crystallizes these connections: it is a beautiful statement tying together Diophantine equations (finding integer or rational solutions) with the analytic properties of L-functions.

2. Millennium Prize Problem
The conjecture is recognized as one of the Clay Mathematics Institute's , highlighting both its importance and its difficulty. A complete proof would be a major milestone in mathematics.

3. Broader Impact
Insights into elliptic curves have already led to breakthroughs in cryptography (e.g., elliptic curve cryptography), factorization methods, and even the proof of Fermat's Last Theorem. A proof or deep progress on BSD would likely spur further advances in these and other fields.

5. Summary

    • Elliptic curves are special cubic curves with a group structure.
    • Rational points on these curves can be finite or infinite in number.
    • The rank measures how many independent ways you can find infinitely many rational points.
    • The L-series of an elliptic curve encodes arithmetic information about the curve.
    • Birch and Swinnerton-Dyer says that how (and how many times) this  vanishes at  exactly matches the curve's rank.


At an undergraduate level, it is enough to grasp the main objects — elliptic curves, ranks, and L-functions — and to understand the conjecture's general claim: there is a precise  signature (the behavior of  at ) that predicts the  property (the rank) of the curve. Proving it would unify these areas in a definitive way and answer a deep question about how rational solutions to these curves are distributed.

Birch and Swinnerton-Dyer conjecture. level: undergraduate. Gemini 1.5 Pro.

$\begin{document}$

$\section*{The Birch and Swinnerton-Dyer Conjecture}$

$\newline$

$\subsection*{Elliptic Curves}$

An elliptic curve $E$ is defined by a Weierstrass equation:

$\begin{equation<em>} y^2 = x^3 + ax + b \end{equation</em>}$

where $a$ and $b$ are integers, and the discriminant $\Delta = -16(4a^3 + 27b^2)$ is non-zero. We are interested in the rational points on $E$ , denoted $E(\mathbb{Q})$ , which are solutions $(x, y)$ where both $x$ and $y$ are rational numbers.

$\newline$

$\subsection*{Group Law}$

Elliptic curves have an addition operation on their points, turning $E(\mathbb{Q})$ into an abelian group.

$\newline$

$\subsection*{Mordell-Weil Theorem}$

The Mordell-Weil theorem states that $E(\mathbb{Q})$ is finitely generated:

$\begin{equation<em>} E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus T \end{equation</em>}$

where $r$ is the $\textbf{rank}$ of $E$ , and $T$ is a finite group (the torsion subgroup).

$\newline$

$\subsection*{L-function}$

The L-function of $E$ , denoted $L(E, s)$ , is a complex-valued function encoding information about the number of points on $E$ over finite fields. It can be analytically continued to the entire complex plane.

$\newline$

$\subsection*{The Conjecture}$

The Birch and Swinnerton-Dyer conjecture connects the rank of $E$ to the behavior of $L(E, s)$ at $s = 1$ :

$\textbf{Conjecture:}$ The order of vanishing of $L(E, s)$ at $s = 1$ is equal to the rank $r$ of the elliptic curve $E$ .

More precisely:

If $L(E, 1) \neq 0$ , then $r = 0$ and $E$ has finitely many rational points.
If $L(E, 1) = 0$ , then $r > 0$ and $E$ has infinitely many rational points. The higher the order of the zero at $s = 1$ , the larger the rank.

$\newline$

$\subsection*{Importance}$

$\textbf{Deep connection:}$ Bridges algebraic and analytic aspects of number theory.
$\textbf{Predictive power:}$ Offers a potential way to determine the rank of $E$ .
$\textbf{Fundamental implications:}$ Connects to modular forms and Galois representations.

$\newline$

$\subsection*{Current Status}$

The conjecture is largely open, but there has been progress:

Proven for curves with rank 0 and 1 (under some conditions).
The Gross-Zagier formula relates $L'(E, 1)$ to the height of a special point on $E$ .

$\newline$

A complete proof remains a major open problem in number theory.

$\end{document}$